Kraus–Dehmer–Schaumann's minimal exponential-entropy graph conjecture

Let $G$ be a graph on $n$ vertices, and let $I_{f^2}(G)$ be the entropy associated with the exponential weight sequence. A generalized star is the tree structure described in the claim, with a central region and branches; its diameter and number of branches are measured in terms of $n$.

Kraus–Dehmer–Schaumann's conjecture. A graph minimizing $I_{f^2}(G)$ for the exponential sequence is a tree. Moreover, it is a generalized star of diameter approximately $\sqrt{2n}$ and therefore has approximately $\sqrt{2n}$ branches.

The conjecture is based on computations for small graph orders, where little was known about minimal entropy graphs. It predicts both the tree structure and the approximate scale of the extremal diameter and branching.